2x2 Normal‑Form Nash Equilibrium Finder

Enter Payoffs (Row, Column)

Cell R1,C1

Row payoff (a₁₁)

Column payoff (b₁₁)

Cell R1,C2

Row payoff (a₁₂)

Column payoff (b₁₂)

Cell R2, C1

Row payoff (a₂₁)

Column payoff (b₂₁)

Cell R2, C2

Row payoff (a₂₂)

Column payoff (b₂₂)

Reset Tip: Positive numbers can represent utilities or payoffs; negatives are allowed.

Game Matrix

Row ↓ / Column →	Column
Row ↓ / Column →	C1	C2
R1	(1.0000, -1.0000) Row BR	(-1.0000, 1.0000) Col BR
R2	(-1.0000, 1.0000) Col BR	(1.0000, -1.0000) Row BR

Best Responses

Row best responses to C1: R1
Row best responses to C2: R2
Column best responses to R1: C2
Column best responses to R2: C1

Dominance

No dominance among row strategies
No dominance among column strategies

Equilibria Summary

Pure equilibria: none

Mixed strategy (interior): Row plays R1 with p = 0.5000 (50.0000%), R2 with 0.5000. Column plays C1 with q = 0.5000 (50.0000%), C2 with 0.5000.

Expected payoffs at mixed equilibrium: Row = 0.0000, Column = 0.0000.

How mixed strategies are computed: Column is indifferent when p satisfies p·b11 + (1−p)·b21 = p·b12 + (1−p)·b22 ⇒ p = (b22 − b21) / (b11 − b21 − b12 + b22). Row is indifferent when q satisfies q·a11 + (1−q)·a12 = q·a21 + (1−q)·a22 ⇒ q = (a22 − a12) / (a11 − a12 − a21 + a22).

If any denominator is zero, either no interior mixed equilibrium exists or there are multiple best replies causing a continuum along an edge.

FAQs

1) What does a pure-strategy Nash equilibrium mean here?

It is a cell where each player’s action is a best response to the other’s action. In the table, such cells are marked with NE.

2) When does a mixed equilibrium appear?

An interior mixed equilibrium exists when the indifference equations yield probabilities between 0 and 1 for both players. If a denominator is zero or probabilities fall outside [0,1], no interior mixed equilibrium exists.

3) What if payoffs tie within a column or row?

Ties create multiple best responses. The tool shows both as best replies and may produce multiple pure equilibria or an entire set along an edge.

4) Do negative numbers work?

Yes. Payoffs can be any real numbers. Only relative differences matter for best responses and mixed strategies.

5) How are dominance statements determined?

A strategy strictly dominates another if it yields a higher payoff in every possible response of the opponent. Weak dominance allows ties in some cases and strict improvement in at least one.

6) Can I use this for coordination or zero-sum games?

Yes. Coordination games typically yield pure equilibria, while zero-sum games like Matching Pennies yield a unique mixed equilibrium; both are handled.

7) Why doesn’t the result match my textbook?

Check that you entered payoffs in the correct cells and that you did not transpose players. Some textbooks normalize or rescale payoffs; such transforms do not change equilibria.